CT measurements and CFD simulations

Chemical Engineering Science 60 (2005) 2215 – 2229 www.elsevier.com/locate/ces

Gas–liquid flow generated by a Rushton turbine in stirred vessel: CARPT/CT measurements and CFD simulations A.R. Khopkara , A.R. Rammohanb , V.V. Ranadea,∗ , M.P. Dudukovicb a Industrial Flow Modeling Group, National Chemical Laboratory, Dr. Homi Bhabha Road, Pune, Maharashtra 411 008, India b Chemical Reaction Engineering Laboratory, Washington University, St. Louis, USA

Received in revised form 15 October 2004; accepted 1 November 2004

Abstract In this work, computer-automated radioactive particle tracking (CARPT), computed tomography (CT) and computational fluid dynamic (CFD) based models were used to investigate gas–liquid flow generated by a Rushton turbine. CARPT and CT measurements were carried out in a gas–liquid stirred vessel operating in two different flow regimes and captured the quantitative Eulerian information of gas–liquid flow. The CARPT data was then used to extract the circulation time distribution in a vessel. A two-fluid model along with the standard k–
turbulence model was used to simulate the dispersed gas–liquid flow in a stirred vessel. Appropriate drag corrections to account for bulk turbulence (along the lines proposed by Brucato et al. (Chem. Eng. Sci. 45(1998) 3295)) were developed to correctly simulate different flow regimes. The computational snapshot approach was used to simulate impeller rotation and was implemented in the commercial CFD code, FLUENT4.5 (of Fluent. Inc., USA). Most model predictions compared favourably with CARPT and CT measurements. Validated CFD models as attempted in this paper are promising to simulation of industrial stirred vessels. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Stirred vessel; Gas–liquid flow; CARPT; CT; CFD

1. Introduction Gas–liquid stirred reactors are widely used in a variety of industrial applications like oxidation, hydrogenation and aerobic fermentation (Shah, 1992). The overall performance of a gas–liquid stirred vessel depends on three key parameters, viz. scale of vessel, impeller shape and dimensions and volumetric gas flow rate (Smith, 1985). These parameters in turn control/affect the gas flow regimes, gas hold-up distribution, power demand, gas–liquid mass transfer coefficient and mixing characteristics in the vessel. Traditionally, empirical correlations are used to estimate prevailing flow regime and corresponding flow characteristics. This empirical information is usually described in an overall/global parametric form and conceals the detailed localized information. The localized information may be crucial for successful design of ∗ Corresponding author. Tel.: +91 20 2589 3400; fax: +91 20 2589 3260.

E-mail address: [email protected] (V.V. Ranade). 0009-2509/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2004.11.044

process equipment (for example, to avoid oxygen depletion in oxidation reaction). Besides this, availability of the relevant empirical information is often a problem either due to the cost or due to the constraints on time. Considering this, it would be most useful to develop computational models which provide quantitative information about the prevailing flow characteristics. Such models have to be validated and for that reason in this work we have experimentally determined the quantitative information of the flow generated by a Rushton turbine for two operating conditions representing two different flow regimes. The measured experimental data was then used to evaluate the developed computational model. Unlike single-phase flows in stirred vessels, there are few experimental studies which report the local information of gas–liquid flow generated by a Rushton turbine (for example, Ranade et al., 2001b; Deen et al., 2002). In most of the studies laser Doppler anemometry (LDA) or particle image velocimetry (PIV) were used. LDA is essentially

2216

A.R. Khopkar et al. / Chemical Engineering Science 60 (2005) 2215 – 2229

a single point measurement technique providing timeaveraged measurements, whereas PIV gives instantaneous whole field information. These optical techniques can be satisfactorily used as long as maximum gas hold-up is less than 5% (Deen et al., 2001). This inherent limitation associated with the LDA/PIV limits their use for measurements in stirred vessels with higher values of gas hold-up particularly in 3–3 cavity regime and flooding regime. In order to overcome this limitation in the present work we have used computer automated radioactive particle tracking (CARPT) and computed tomography (CT) for obtaining quantitative information about 3–3 cavity and flooding regimes. CARPT and CT can be used for flow measurements in optically opaque and non-transparent multiphase systems (e.g. bubble columns operated at high gas velocities and gas hold-ups in excess of 25% (Devnathan et al., 1990; Kumar et al., 1997; Dudukovic, 2002). Recently, Rammohan et al. (2001a,b) successfully used CARPT to extract Eulerian as well as Lagrangian information of single-phase flow in stirred vessel. In this work, we report application of CARPT for characterizing gas–liquid flow generated by a Rushton turbine in a fully baffled vessel. Apart from providing Eulerian information, CARPT provides unique Lagrangian information. The CT was used to measure the local values of gas hold-up in the vessel. The measured liquid velocities and local gas hold-up values were then used to validate the computational models based on computational fluid dynamics (CFD). Several attempts have been made in recent years to develop computational models of gas–liquid flows in stirred vessels (for example, Gosman et al., 1992; Morud and Hjertager, 1996; Bakker and van den Akker, 1994; Ranade and van den Akker, 1994; Ranade and Deshpande, 1999; Lane et al., 1999, 2000; Khopkar et al., 2003). Most of these studies were restricted to very low gas flow rates with complete dispersion regime present in the reactor. The 3–3 cavity regime (loading condition) is however one of the most commonly used flow regime in practice. Simulations at higher gas flow rates resulting in 3–3 cavity regime or flooding regime are rare due to unavailability of experimental data. In this work, we made an attempt to develop a CFD model to capture these flow regimes. The computational snapshot approach (Ranade, 2002) was used to simulate impeller rotation. Turbulent dispersed two-phase flow in stirred vessels was simulated using a two-fluid model with the standard k–
turbulence model. The model predictions of these flow regimes were evaluated by comparison with the measured experimental data obtained using CARPT and CT for the first time. The computational model was then used to understand the effect of high gas loading on the flow near impeller blades. The particle trajectory simulations were carried out to obtain the Lagrangian information. The CFD models were also used for the first time to predict Lagrangian information obtained from CARPT for gas–liquid stirred reactors. The details of the experimental method and the computational model,

as well as the results obtained are discussed in following sections. 2. Experimental apparatus and method CARPT and CT measurements of the three-dimensional velocity components and local gas hold-up values were carried out in a flat bottom fully baffled cylindrical vessel (of diameter, T =0.2 m, height, H =0.2 m). Four baffles of width T /12.5 were mounted perpendicular to the vessel wall. The shaft of the impeller (ds = 0.006 m) was concentric with the axis of vessel and extended to the bottom of the vessel. A six-bladed Rushton turbine (of diameter, Di = T /3, impeller blade height, W = Di /4 and impeller blade width, B = Di /5) was used during the CARPT and CT measurements. The impeller off-bottom clearance was (C = T /3) measured from the center of the impeller disc to vessel bottom. The rotational speed of the impeller was set to 200 rpm (F r = 0.0755), which is equivalent to an impeller Reynolds number (Re) of 14,800. Gas was introduced into the vessel through a ring sparger of diameter 0.0666 m. The ring sparger was located at T /6 from the bottom of tank and has an outer diameter equal to impeller diameter. When gas is sparged in a stirred vessel, depending on relative strengths of impeller generated flow and gas generated flow, different flow regimes are realized. Several studies in the past have investigated these flow regimes. The flow regime map emerges from these studies (see for example, Chapman et al., 1983; Warmoeskerken and Smith, 1985; Nienow et al., 1985, 1988). These flow regime maps, as well as visual observations, were used to identify the flow regimes. In the present study, experiments were carried out for two volumetric gas flow rates corresponding to gas flow numbers of 0.042 and 0.084. The values of these two gas flow rates were chosen in such a way that the prevailing flow regimes in the stirred vessel are the 3–3 cavity and flooding (ragged cavity) regimes. All the experiments were carried out with tap water as the liquid phase and air as the gas phase. 2.1. CARPT technique and setup The CARPT technique involves following (tracking) the motion of a neutrally buoyant, with respect to the liquid, radioactive tracer with the help of Sodium Iodide scintillation detectors. The tracer used in the present study was a 1.0 mm polypropylene spherical bead with radioactive Sc-46 embedded in it (strength 100
Ci). During the experiments the position of the particle was determined by an array of 16 scintillation detectors mounted on Aluminium supports, which are arranged on an octagonal base (see Fig. 1). The scintillation detectors recorded the radiation intensity emitted by the tracer. The radiation intensity recorded by each detector decreases with increasing distance between the detector and particle. Only the photo-peak fraction of the total energy spectrum was acquired to improve the accuracy of

A.R. Khopkar et al. / Chemical Engineering Science 60 (2005) 2215 – 2229

2217

Fig. 1. Schematic of the CARPT setup for the stirred vessel. (a) Front view, (b) top view.

the CARPT reconstruction method (for further details refer to Rammohan, 2002). Typically for a given impeller speed and gas sparging rate, particle trajectory information was collected over a period of 24 h with a sampling frequency of 200 Hz (however data was reprocessed at 50 Hz for velocity calculations in view of the dynamic bias reported by Rammohan et al. (2001a)). In order to estimate the position of the particle from the acquired radiation intensities, a calibration of CARPT technique was carried out. For details related to calibration of CARPT technique, error analysis and post-processing of CARPT data refer to Rammohan et al. (2001a).

2.2. CT technique and setup The time averaged cross-sectional gas hold-up distribution at three axial locations (Z/T = 0.25, 0.5 and 0.75) were obtained with CT. The same stirred vessel setup used for CARPT measurements was used for CT measurements. Fig. 2 shows the schematic diagram and top view of the CT setup. The support structure of the CT setup consists of four threaded vertical guide rods on which a perfectly horizontal plate was positioned so as to allow its smooth vertical motion automated by gears. On this a gantry plate was fixed. The gantry plate houses the 100 mCi lead shielded Cs-137 source and an array of seven NaI (Tl) detectors, which were positioned across the vessel diameter facing the source. The whole assembly (source plus detector array) rotated around the vessel during the data acquisition process. Thick lead shielding in front of the source was used to collimate the radiation into a fan beam whose angle can be varied to envelop the vessel. The detector array also has a lead collimator that can move, controlled by a stepper motor, in front of the detector (Fig. 2b). This modification proposed by Kumar et al. (1995) has been used to acquire 14 independent projection measurements per detector. The collimator slots have a

width of 3.0 mm. During the data acquisition process, with the source switched on, the stepper motors control the motion of the rotating gantry in 99 positions spanning the 360◦ around the vessel. For each such “view”, the detector collimator moved 14 steps (and was then brought back to its original position) so as to acquire 14 independent projections for each detector per view. Therefore seven detectors were sufficient to span the fan beam and total of 9702 projections (14 × 7 × 99) were acquired during measurements. The total scanning time (over which the holdup distribution was averaged) was a little over 3 h. The photon count data was acquired using a threshold of 420 mV which ensured that only the photo-peak photons were collected and not the entire spectrum. For image reconstruction the estimation maximization algorithm was used (for further details refer to Kumar et al. (1995) and Kumar and Dudukovic (1997)). The tomography problem is an inverse problem of reconstructing the total effective attenuation function ϑeff (x, y) based on measurements of the lineaveraged attenuations along various chords in the system of interest. These line averaged attenuations or projection data (as defined in the CT literature) were obtained all around the column and at various chords. From these measurements the time averaged effective attenuation coefficient (ϑeff (x, y)) was obtained in every pixel by reconstructing the image from the entire set of projection measurements. In order to ensure equivalent convergence of the different reconstruction cases (corresponding to changes in impeller speeds and gas flow rates) an internal convergence criterion was defined as: Npix

Re sN+1 =


i=1

2 (N+1 − N i ) , i

(1)

where Re sN+1 is the residual at the N + 1th iteration, N+1 i is the value of the holdup in the ith pixel at the N + 1th iteration. Typically the residues were required to be smaller than 0.01–0.0001.

2218

A.R. Khopkar et al. / Chemical Engineering Science 60 (2005) 2215 – 2229

Fig. 2. Schematic of the CT scanner with the stirred vessel. (a) Front view, (b) top view, at one specific location of gantry plate (note that dimensions and angles are not to scale).

3. Mathematical modeling 3.1. Transport equations For simulating gas–liquid flow in stirred vessels, a twofluid model based on Eulerian–Eulerian approach was used in this work. The mass and momentum balance equations for each phase may be written as:

j(q q ) j + (q q Uqi ) = 0, jt jxi j(q q Uqi ) j (q q Uqi Uqj ) + jt jx i    jUqj jUqi jp j + q q + = −q jxi jxi jx jxi  j  jUqm 2 j + q q gi + Fqi − , q q 3 jx i jxm

(2)

(3)

where Uqi is velocity of phase q in i direction and Fqi is inter-phase momentum exchange terms. Experiments were carried out at impeller Reynolds number of 14,800. Previous studies indicate that when impeller Reynolds number is greater than 10,000 turbulence is fully established. Even at smaller impeller Reynolds number, predictions with standard k–
model (which is mainly applicable for fully developed turbulent flow) were found to be reasonable (see for example, estimation of impeller pumping number at different Reynolds number reported by Ranade (1997)). Therefore in this work, the standard k–
model of turbulence along with the Reynolds averaged governing

equations were used for simulating turbulent gas–liquid flows. For more details of time averaged two-phase balance equations, the reader is referred to Ranade (2002). It may be noted that, while writing the time averaged mass balance, turbulent dispersion of dispersed phase was not considered. The numerical study of Khopkar et al. (2003) indicated that the turbulent dispersion terms were significant only in the impeller discharge stream. Even near the impeller, the influence of dispersion terms on predicted results was not quantitatively significant (difference was less than 5%). Therefore, turbulent dispersion of dispersed phase was not considered while formulating time-averaged Equation (3). The governing equations for turbulent kinetic energy, k and turbulent energy dissipation rate,
, were solved only for the liquid phase and are listed below:

j j ( l k) + ( l Uli k) jt l jxi l   j  jk + l (G − l
), l t = jxi  k j xi j j ( l Uli
) (l l
) + jx i l jt    j t j

 = + l C1 G − C2 l
, l jxi 
jx i k

(4)

(5)

where G is turbulence generation rate and t is turbulent viscosity: G = 21 t (∇ U¯ + (∇ U¯ )T )2 ,

t =

l C k 2 .


(6)

A.R. Khopkar et al. / Chemical Engineering Science 60 (2005) 2215 – 2229

2219

The standard values of k–
model parameters were used in present simulations. The inter-phase momentum exchange term, Fqi consists of four terms: the Basset force, the virtual mass force, the lift force and the interphase drag force. In the bulk region of the vessel, the velocity gradients are not large. Near the impeller, pressure gradients and interphase drag forces mainly dominate the motion of the bubbles. An order of magnitude analysis indicates that the magnitude of the lift force is much smaller than the interphase drag force. Recent numerical experiments reported by Khopkar et al. (2003) indicate that the effect of the virtual mass force is not significant in the bulk region of stirred vessel. In most cases, the magnitude of the Basset force is also much smaller than that of the inter-phase drag force. Therefore, to reduce computational costs, only the inter-phase drag force was considered in the inter-phase momentum exchange term. The inter-phase drag force exerted on phase 2 in i direction is given by: Fqi = FD2i = −

3 1 2 1 C D

 0.5 (U2i − U1i )2 (U2i − U1i ) . (7) 4db

In gas–liquid stirred vessels the interphase drag coefficient, CD , is a complex function of the drag coefficient in a stagnant liquid, the gas hold-up and prevailing turbulence. Recently, Lane et al. (2000) studied the effect of turbulence on the drag coefficient (slip velocity). Based on a comparison of the predicted gas volume fraction distribution with the experimental data, they recommended a turbulence correction factor proposed by Brucato et al. (1998) but with a lower value of the correlation constant. Following this, we have used the following correlation for calculation of the drag coefficient:  3 CD − CD0 db =K , CD0     

2.667Eo 24  1 + 0.15Re0.687 , , CD0 = max b Eo + 4.0 Reb (8) where  is the Kolmogorov length scale, db is the bubble diameter and K is an empirical constant, which was set to 6.5 × 10−6 . Eq. (8) thus accounts for the increased drag coefficient due to turbulence. The gas–liquid flow in the stirred vessel was simulated using the computational snapshot approach. In this approach, the impeller blades are considered as fixed at one particular position (similar to taking a snapshot of the rotating impeller) with respect to the baffles. Recently Ranade (2002) discussed the development of the snapshot approach in detail and therefore it is not repeated here. The flow is simulated for a specific blade position with respect to the baffles. The predicted results are like a snapshot of flow for a specific blade position with respect to baffles and do not change with simulation time after satisfactory convergence criteria

Grid Details

:

r×θ×z

: 58 × 95 × 64

Impeller blade

: 14 × 3 × 18

Inner region

: 12 ≤ k ≤ 53 j ≤ 42

Fig. 3. Computational grid and solution domain.

was achieved. The results obtained with a specific snapshot position were not found to be significantly different from the ensemble average of a number of snapshots (Ranade and van den Akker, 1994). The computational snapshot approach was implemented in the commercial CFD code FLUENT 4.5 (of Fluent Inc., USA) using user-defined subroutines. 3.2. Solution domain and boundary conditions Considering the symmetry of geometry, half of the vessel was considered as a solution domain (see Fig. 3). The baffles were considered at angles of 45◦ and 135◦ . The impeller was positioned in such a way that three blades were located at angles of 30◦ , 90◦ and 150◦ . As discussed by Ranade (2002), the computational snapshot approach divides the solution domain into an inner region, in which time derivative terms are approximated using spatial derivatives and an outer region, in which time derivative terms are neglected. The boundary between the inner and outer regions needs

2220

A.R. Khopkar et al. / Chemical Engineering Science 60 (2005) 2215 – 2229

to be selected in such a way that the predicted results are not sensitive to its actual location. In the present work, for all simulations, the boundary of the inner region was positioned at r = 0.06 m and 0.046 m
z
0.133 m (where z is the axial distance from the bottom of the vessel). In the present work, the sparger was modeled as a solid wall. The mass source of the gas phase was specified one cell above the sparger wall to simulate gas introduction into the vessel. Special boundary conditions are needed to simulate the gas–liquid interface at the top through which bubbles escape the solution domain. Recently, Ranade (2002) discussed different possible approaches to treat gas–liquid interfaces in detail. We have modeled the top surface of the dispersion as velocity inlet. The outgoing (axial) velocity of gas bubbles was set equal to the terminal rise velocity of gas bubbles (estimated as 0.2 m/s for air bubbles). All the other velocity components for gas and liquid phase were set to zero. Implicit assumption here is that gas bubbles escape the dispersion with terminal rise velocity. Since the liquid velocity near the top gas–liquid interface is small and the overall volume fraction of gas is also small (< 5%), this assumption is reasonable. It should be noted that even after defining the top surface as an inlet, gas volume fraction at the top surface is a free variable. The mass and momentum conservation equations for the gas phase were solved and the gas distribution within the vessel was predicted. Gas mass conservation was verified by comparing the integral gas mass flow rate across various horizontal planes with the input gas mass flow rate at the sparger. In a gas–liquid stirred vessel, there may be a wide distribution of bubble sizes. The prevailing bubble size distribution in a gas–liquid stirred vessel is controlled by several parameters like vessel and sparger configuration, impeller speed and gas flow rate. It is possible to develop a detailed multi-fluid computational model using the population balance framework to account for bubble size distribution. We have developed, applied and validated such models for gas–liquid flow in bubble columns (Buwa and Ranade, 2002; Chen et al., 2004, 2005a,b; Rafique et al., 2004). However, use of multi-fluid models based on population balances increases the computational demands by many fold. Unfortunately available experimental data of bubble size distribution in stirred vessel is not adequate to calculate the parameters appearing in coalescence and break-up kernels. Apart from the uncertainty in parameters of coalescence and break-up kernels, there is significant uncertainty in estimation of interphase drag force on gas bubbles in presence of other bubbles and high levels of turbulence prevailing in the vessel. Considering these issues and the present state of understanding, the option of using a multi-fluid model for stirred vessels is premature especially since we have not measured the bubble size distribution in our experimental stirred vessel. Barigou and Greaves (1992) reported experimentally measured bubble size distribution for the stirred vessel of 1 m diameter (of size T = H = 1 m, C = T /4 and Di = T /3). Their experimental data clearly indicates that the bubble sizes in the

bulk region of the vessel vary between 3.5 and 4.5 mm at high gas flow rate. Considering these issues, in the present work, a single bubble size was specified (4 mm) for all the simulations. Fluid properties were set to those of water and air for the primary and secondary phases, respectively. The mass and momentum conservation equations for the gas phase were solved and the gas distribution within the vessel was predicted. Mass conservation was verified by comparing the integral gas mass flow rate across various horizontal planes with the input gas mass flow rate at the sparger. It is very important to use an adequate number of computational cells while numerically solving the governing equations over the solution domain. The prediction of turbulence quantities is especially sensitive to the number of grid nodes and grid distribution within the solution domain. Our previous work (Ranade et al., 2001a) as well as other published work (for example, Ng et al., 1998; Wechsler et al., 1999) gives adequate information to understand the influence of the number of nodes on the predicted results. It was demonstrated that in order to capture the details of flow near impeller, it is necessary to use at least 200 grid nodes to resolve the blade surface. Based on previous experience and some preliminary numerical experiments, the numerical simulations for gas–liquid flows in stirred vessel were carried out for grid size (r ×  × z: 58 × 95 × 64). In the present work we have used (r ×  × z: 14 × 3 × 18) grid nodes covering the impeller blade. The boundary of the inner region was positioned at j
42 and 12
k
53 (where j is cell number in radial direction from shaft and k is cell number in axial direction from bottom of vessel). The computational grid used in the present work is shown in Fig. 3. In the present work, QUICK discretization scheme with SUPERBEE limiter function (to avoid non-physical oscillations) was used. Standard wall functions were used to specify wall boundary conditions. The computational results are discussed in the following section.

4. Results and discussion 4.1. Bulk flow characteristics 4.1.1. Liquid phase velocities The gas–liquid flows generated by the Rushton turbine in the stirred vessel were simulated for two gas flow numbers 0.042 and 0.084 (corresponding to a volumetric gas flow rates (Qg ) of 4.17 × 10−5 and 8.33 × 10−5 m3 /s, respectively) and for a Froude number equal to 0.0755 (corresponding to an impeller rotational speed of 200 rpm). Under these operating conditions, the fluid dynamics in the vessel represents the 3–3 cavity and the flooding regime. The predicted liquid phase velocity vectors for both operating conditions are shown in Figs. 4a and 5a. It can be seen from Figs. 4a and 5a that the computational model captured the well known two-loop structure generated by the Rushton turbine. The

A.R. Khopkar et al. / Chemical Engineering Science 60 (2005) 2215 – 2229

2221

0.6 m/s 0.3 m/s

0 m/s

(a)

(b) Legends:

0 m/s

(c) Experimental data Predicted results

Fig. 4. Comparison of predicted results and experimental data for gas–liquid flow (3–3 cavity regime), F l = 0.042, F r = 0.0755 and Utip = 0.7 m/s: (a) simulated mean liquid flow field: mid-baffle plane; (b) radial velocity; and (c) tangential velocity.

0.6 m/s 0.3 m/s

0 m/s

(a)

(b) Legends:

0 m/s

(c) Experimental data Predicted results

Fig. 5. Comparison of predicted results and experimental data for gas–liquid flow (flooding regime), F l = 0.084, F r = 0.0755 and Utip = 0.7 m/s: (a) simulated mean liquid flow field: mid-baffle plane; (b) radial velocity; and (c) tangential velocity.

2222

A.R. Khopkar et al. / Chemical Engineering Science 60 (2005) 2215 – 2229

(b)

(a)

Contour level: ≤ 0.5%

≥ 5.0%

(Gas volume fraction: 10 uniform contours between 0.5 to 5%)

Fig. 6. Simulated gas hold-up distribution at the r–z plane for gas–liquid flow, F r = 0.0755 and Utip = 0.7 m/s: (a) 3-3 cavity regime, Fl = 0.042: mid-baffle plane; (b) flooding regime, Fl = 0.084: mid-baffle plane.

predicted velocity fields for both regimes show the upward inclination of the impeller discharge stream. To understand the contribution of the rising gas on the upward inclination of the impeller discharge stream, the predicted velocity fields for aerated conditions were compared with the predicted velocity field for single-phase flow (not shown here for brevity). It was observed that increase in gas flow rate increases upward inclination of impeller discharge stream. Apart from the increased upward inclination of impeller discharge stream, the predicted velocity fields for aerated conditions show the presence of secondary circulation loops above the impeller. Such secondary loops are some times observed with low impeller Reynolds numbers single-phase flows. However, in the present case (impeller Reynolds number of 14,800) no such secondary circulation loop was observed for the case of single-phase simulation. Recent experimental as well as numerical study of Hartmann et al. (2004) also confirmed the absence of secondary circulation loop for single-phase flow for even smaller value of Reynolds number (7300). The predicted gas flow fields were analysed to understand the possible reasons for the generation of secondary circulation loops near the top surface. The predicted gas flow field (see Fig. 6 for predicted gas hold-up distribution) shows the inward motion of the gas phase near the top surface of liquid. This inward movement of upward rising gas generates the secondary circulation of liquid in the upper region of the vessel. The observed secondary loop was more prominent in the flooding regime (Fig. 5a). Quantitative comparison of predicted values of the liquid phase radial and tangential velocities with experimental data for 3–3 cavity regime is shown in Fig. 4b and c. The predicted values of radial velocities (axial profiles plot-

ted at three radial distances, 0.025, 0.0375 and 0.0625 m, measured from center axis of vessel) show good agreement with the experimental data in Fig. 4b. The computational model has over predicted the radial velocity values in impeller discharge stream. However, low values of radial velocities were predicted near the top surface of liquid. The presence of secondary circulation loop near the top surface was clearly seen in the axial profile of the radial velocity plotted at radial distance of 0.0625 m measured from central axis of the vessel. The comparison between simulated axial profile of radial velocity and experimental data shows that the predicted secondary circulation loop was bigger in size than the circulation loop observed during experiments. The comparison of predicted values of tangential velocity with experimental data at first two radial distances is shown in Fig. 4c. The comparison of the predicted results with the experimentally measured data was reasonably good in the bulk region of the vessel. Quantitative comparison of the predicted values for the liquid phase radial and tangential velocities with experimental data in the flooding regime is shown in Fig. 5b and c. The comparison of axial profiles of radial velocity at three radial distances in Fig. 5b was reasonably good. Higher values of radial velocities were predicted in the impeller discharge stream. The computational model successfully captured the secondary circulation loop near the top surface of the liquid (see Fig. 5b). However, the predicted size of the secondary circulation loop was bigger than the circulation loop observed in the experimental measurements. The comparison of predicted values of tangential velocity with experimental data is shown in Fig. 5c. The computational model has over-predicted the tangential velocity values in impeller discharge stream. The comparison of predicted values of tangential velocity with experimental data in the bulk volume of reactor was reasonable. Predicted influence of gas flow rate on gross characteristics like power and pumping numbers are also of interest. Pumping and power numbers were calculated from simulated results as:  B/2  2 −B/2 0 l ri Ur d dz NQ = , (9) N D 3i  2 l 
dV NP = V , (10) N 3 Di5 where B is blade height, Di is impeller diameter, N is impeller speed, ri is impeller radius and Ur is radial velocity. The calculated values of pumping and power number from the simulated results are listed in Table 1. The simulations captured the decrease in impeller pumping as well as power dissipation, with increase in gas flow rate. The predicted value of pumping number for single-phase flow (0.65) was in good agreement with the experimental value (0.6) reported by Ranade et al. (2001b). It can be seen that the predicted power dissipation for the flooding regime is higher than that

A.R. Khopkar et al. / Chemical Engineering Science 60 (2005) 2215 – 2229

2223

Table 1 Gross characteristics of gas–liquid flow generated by a Rushton turbine Operating conditions

Single-phase flow 3–3 flow regime (F l = 0.042 & F r = 0.0755) Flooding regime (F l = 0.084 & F r = 0.0755)

Total gas hold-up (%)

Average circulation time (s)

Predicted results

Predicted

Experimental

Predicted

Experimental

N Pg

N Qg

NPg /NP

— 1.52 1.94

— 1.4 1.6

6.35 7.34 7.95

— 5.29 5.99

4.67 3.5 3.69

0.65 0.58 0.53

1 0.75 0.79

for the 3–3 cavity regime. Similar increase in power number for flooding regime compared with power number for 3–3 cavity regime was reported in previous studies (for example, Nienow et al., 1977). Overall, reasonably good agreement between the experimentally measured time averaged CARPT data and simulated results were observed for both 3–3 cavity and flooding regime. The major disagreement was observed in impeller discharge stream and near the top surface. In both cases, the computational model over-predicted the velocity values and the size of the secondary circulation loop. The computational grid, turbulence model, use of a single bubble size and inadequacies of inter-phase momentum exchange term are some of the possible reasons for the observed disagreement. Since we have used number of computational cells more than 350 k and used higher order discretization scheme, the contribution of computational grid in the observed disagreement might be low. One of the key reasons of the observed disagreement might be inaccurate estimation of inter-phase drag force. Knowledge about influence of bubble size, neighbouring bubbles and prevailing turbulence on inter-phase drag force is not adequate. Moreover, the prevailing levels of turbulence were estimated using the standard k–
model of turbulence. In absence of better understanding, single-phase parameters were used and influence of bubbles on turbulence generation was ignored. The gas–liquid flow near impeller blades involves complex interaction of bubbles, gas cavities and trailing vortices behind the blades. Further studies throwing light on such complex interactions are needed for improving computational models of gas–liquid flow in stirred vessels. The top surface of gas–liquid dispersion was assumed to be flat in the present work. This assumption may have an adverse impact on motion of gas bubbles near top surface and may lead to inaccuracies in prediction of secondary circulation loop. Further numerical experiments with different implementation of top boundary conditions, as discussed by Ranade (2002), will be useful to gain further understanding of possible ways of enhancing the accuracy of computational models. The CARPT method cannot give the detailed information of flow around impeller blades. Such information is useful to understand the performance of impeller in an aerated stirred vessel. The computational model was then used to understand the flow around impeller blades. Before discussing the flow around impeller blades the gas hold-up distribution in

the vessel was studied using CT measurements and computational model. 4.1.2. Gas hold-up distribution The gas hold-up distribution in the vessel is strongly affected by the gas and liquid properties, prevailing flow regime and reactor internals. In the present work, we have used the computational model and CT technique to study the gas hold-up distribution in 3–3 cavity and flooding regime. The simulated gas hold-up distributions for both operating conditions (F l = 0.042 and F r = 0.0755 (3–3 cavity) and F l = 0.084 and F r = 0.0755 (flooding)) are shown in Fig. 6a and b. The simulations captured the inefficient dispersion of gas for the flooding regime. The predicted contour plots clearly show inward movement of gas while rising through the vessel. This inward movement of gas generates the secondary circulation loop at the upper part of the vessel. The observed inward movement of the gas was seen to be very strong in the flooding regime (see Fig. 5a) in comparison with 3–3 cavity flow regime (see Fig. 4a). The comparison of predicted values of gas hold-up with the experimental data, at two axial locations (0.1 and 0.15 m from bottom of tank), is shown in Fig. 7a and b for the flooding regime. Unlike the CFD predictions, which show significant inward (towards shaft) movement of gas bubbles, the CT results indicate more volume fraction of gas near vessel wall. The CT results were found to be sensitive with respect to the convergence criterion used during the processing of raw data (Eq. (1)). The details of the sensitivity of the processed results with respect to different parameters may be obtained from the authors. The smaller the value of convergence criterion, the processed contours of the gas volume fraction showed increasing discontinuities for reasons currently poorly understood. The use of a threshold of 420 mV, which ensured that only the photopeak photons were collected, use of a technique, which captures multiple projections per view causing inadequate number of photon counts, or inadequate averaging time, are some of the possible reasons. It should be noted that preliminary CT measurements were carried out to ensure that basic reconstruction methodologies in absence of gas are adequately accurate (by comparing reconstructed shapes with stationary objects). The entire CT measurement procedure for gas–liquid flow in stirred vessel is being scrutinized to develop ways to overcome some of the observed limitations.

2224

A.R. Khopkar et al. / Chemical Engineering Science 60 (2005) 2215 – 2229

(i) CT measurement

(ii) Predicted

(a)

(i) CT measurement

(ii) Predicted

(b) Contour level: ≤ 0.5%

≥ 10%

(Gas volume fraction: 10 uniform contours between 0.5 to 10%)

Fig. 7. Comparison of predicted gas hold-up distribution at the r– plane for gas–liquid flow, F l = 0.084, F r = 0.0755 and Utip = 0.7 m/s: (a) gas holdup distribution at r- plane, 0.1 m from bottom of vessel; and (b) gas holdup distribution at the r- plane, 0.15 m from bottom of vessel.

4.2. Flow around impeller blades The overall performance of an impeller in a gas–liquid stirred vessel is greatly influenced by the flow around impeller blades: mainly structure of trailing vortices, lowpressure region and the gas accumulation behind impeller blades. The trailing vortices and the low-pressure region associated with the vortex core control the extent of gas accumulation behind impeller blades. This gas accumulation commonly known as gas cavities significantly reduces the pumping efficiency of an impeller and power dissipated by an impeller. Therefore, it is necessary to study the flow around impeller blades to understand performance of impeller blades in gas–liquid systems. There have been few experimental attempts to study the flow around impeller blades (for example, Deen et al., 2002; Khopkar et al., 2003). In both studies PIV measurement technique was used to study the flow field around impeller blades. But the constraint associated with PIV measurements limits the use of experimental techniques to low gas flow rates (low values of gas hold-up). For 3–3 cavity and flooding regimes large cavities were observed behind impeller

blades. In these regimes the conventional optically based experimental techniques fail because of high values of gas hold-up around impeller blades. In this work we have used our computational model to understand the flow around the impeller blades. The predicted flow characteristics near the impeller blades are shown in Figs. 8 and 9 in the form of iso-surfaces of gas volume fraction; contours of gas volume fraction, z-vorticity and turbulent kinetic energy dissipation rate for two flow conditions. It can be seen from Fig. 8 that the computational model successfully captured the gas accumulation in the lowpressure region behind impeller blades. The sparged gas interacts with the trailing vortices and gets accumulated in the low-pressure region associated with the vortices. The contours and iso-surface of gas volume fractions indicate that, a large portion of the gas in the impeller swept region gets accumulated in the low-pressure region associated with the lower trailing vortex. The examination of flow and gas accumulation around impeller blades provides clues regarding the prevailing flow regime. The iso-surfaces of gas volume fraction (iso-value = 0.12) shown for two flow regimes clearly indicate the progressive increase in the quantity of gas accumulated behind the impeller blades as one progresses from the 3–3 regime to the flooding regime. The computational model could not capture the asymmetry in cavity shape for the 3–3 cavity flow regime. The physical reasons for possible asymmetry in cavity shapes are not yet clear and further work is needed to capture alternating cavity structure/gas accumulation behind impeller blades. The predicted results indicate that the computational model reasonably captured the influence of the flow regime on gas accumulation behind impeller blades. The predicted contour plots of z-vorticity and turbulent dissipation rate around impeller blades are shown in Fig. 9a and b, respectively. The accumulated gas disrupts the shape of the lower trailing vortex (Fig. 9a). The disruption of the lower trailing vortex was more prominent in the flooding regime because of high gas accumulation (see Fig. 8b-ii). Khopkar et al. (2003) have also reported the disruption of the trailing vortex due to high gas accumulation. Similar behaviour was not observed for the upper vortex in both the regimes because of low gas accumulation. The predicted contours of z-vorticity for flooding regime clearly show the upward inclination of lower trailing vortex. The upward inclination of lower trailing vortex was due to the buoyancy of the gas pocket present in the vortex core. The predicted contours of dissipation rates show significant increase in the dissipation as the flow regime changes from 3–3 cavity to flooding regime. The high gas accumulation and disruption of trailing vortex may be a possible reasons for the prediction of higher values of dissipation rate for the flooding regime. The increase in predicted values of dissipation rate around impeller blades was the reason for prediction of the higher value of power number in flooding regime compared with 3–3 cavity regime.

A.R. Khopkar et al. / Chemical Engineering Science 60 (2005) 2215 – 2229

(i) 3-3 Cavity regime, Fl = 0.042

(ii) Flooding regime, Fl = 0.084

(i) 3-3 Cavity regime, Fl = 0.042

(ii) Flooding regime, Fl = 0.084

2225

(a)

Contour level: ≥30%

0% (b)

Fig. 8. Predicted gas hold-up distribution behind impeller blades, F r = 0.0755 and Utip = 0.7 m/s: (a) iso-surface of gas volume fraction: iso-value = 0.12; and (b) gas accumulation behind impeller blades (impeller is moving in anti-clockwise direction).

(i) 3-3 Cavity regime, Fl = 0.042

(ii) Flooding regime, Fl = 0.084

Contour level: ≤-200 s-1

≥ 200 s-1

(a)

(i) 3-3 Cavity regime, Fl = 0.042

(ii) Flooding regime, Fl = 0.084

Contour level: 0 m2/s3

≥1 m2/s3

(b) Fig. 9. Details of predicted flow around impeller blades, F r = 0.0755 and Utip = 0.7 m/s: (a) vorticity (z-direction, 10 uniform contours between −200 and 200/s); and (b) turbulent kinetic energy dissipation rate (10 uniform contours between 0 and 1 m2 /s3 ): impeller is moving in anti-clockwise direction.

2226

A.R. Khopkar et al. / Chemical Engineering Science 60 (2005) 2215 – 2229

4.3. Circulation time distribution The CARPT technique provides unique Lagrangian information. Such information can provide insight into circulation patterns and mixing in the stirred vessel. For example, Rammohan et al. (2001a) used CARPT data to calculate the Sojourn time distribution in a stirred vessel to quantify the dead zones present in the vessel. In the present work, we have used the CARPT data to quantify the circulation time distribution in the stirred vessel and to investigate the influence of gas flow on this distribution. The circulation time was defined as the time required by a tracer particle to return to impeller swept volume. If particle was found to return to impeller swept volume in less than 0.1 s, the event was not counted to avoid situations representing local re-circulation. The distribution of the times taken by the particle to visit the impeller swept region was calculated and the average circulation time was estimated from this distribution. The circulation time distributions obtained from the CARPT data (12,230 circulations for 3–3 cavity regime, and 20,000 circulations for flooding regime) are shown in Fig. 10. It can be seen that there are very few re-circulations which require more than thrice of average circulation time. The experimentally measured values of the average circulation time are listed in Table 1. It is instructive to simulate the circulation time distribution from the CFD model and to compare these results with the experimental data. Using the Eulerian flow field obtained as discussed in previous subsections, the particle trajectories were simulated. These particle trajectory simulations were carried out for three gas flow numbers (F l = 0.0, 0.042 and 0.084) and constant impeller speed (F r = 0.0755). A single neutrally buoyant particle (same density as water) of diameter 1 mm (same as that used in the experiments) was released in the liquid for particle trajectory calculation. The particle was released in the liquid phase at seven different positions in the solution domain. These seven locations were selected at random. The motion of the particle in the liquid phase was simulated in a Lagrangian framework. The details of the trajectory calculations are recently discussed in Rammohan et al. (2003) and hence not repeated here. The simulated particle trajectories were used to calculate the circulation time distribution. The simulated circulation time distribution is compared with the CARPT data in Fig. 10. It should be noted that the simulated circulation time distribution was calculated based on 1000 circulations of the particle unlike more than 12,000 circulations in the CARPT data. It can be seen that the predicted circulation time distribution agrees reasonably well with the CARPT data for both the regimes. The values of average circulation time were calculated from the distribution of circulation time. The predicted and experimentally measured values of average circulation time are listed in Table 1. The computational model has captured the increase in circulation time with increase in gas flow rate. The predicted values of average circulation time were higher than the experimental data for both flow conditions.

The over prediction of the number of long time circulations is possibly the reason for over prediction of average circulation time. The distributions generated by the model based on the time averaged flow data may not have converged to the experimentally determined distributions. It is also possible that because of inadequate turbulent dispersion, some of the simulated particle trajectories are taking much longer time leading to over prediction of circulation time. Further studies of the role of turbulent dispersion and total number of trajectories on predicted circulation time are needed to resolve the observed disagreement. The developed computational model not only captured the essential features of the gas–liquid flows operating in different flow regimes but also predicted some of the Lagrangian information of flow. Such validated model will be useful to understand the implications of scale-up and scaledown on fluid dynamics and ultimately reactor performance. For gas–liquid flows, bubble size remains more or less the same during scale-up or scale-down while the rest of the key scales like impeller blade width/diameter and vessel diameter change significantly. The interaction of bubbles with impeller blades (and trailing vortices behind them) therefore gets significantly influenced by the scale-up or scaledown operation. The validated computational models can provide useful information on how bubble/impeller interaction changes with scale. Such interactions influence two important design parameters, namely power dissipation and pumping capacity of the impeller. Circulation time distributions also dramatically change during scale-up and scaledown operations. The model and the results presented here provide a useful basis for obtaining such crucial information needed for practice.

5. Conclusions In this work, CARPT and CT measurements and CFD models were used to study the gas–liquid flow (3–3 cavity and flooding regime) generated by a standard Rushton turbine in a stirred vessel. The CARPT technique quantitatively captured the Eulerian liquid flow characteristics. In addition the CARPT technique provided valuable Lagrangian information of the liquid flow. Some difficulties in CT measurements of gas phase hold-up distributions in the stirred vessels were identified. The mean flow and turbulent characteristics were computed by solving the time-averaged two-fluid model equations with the standard k–
turbulence model for two gas flow rates (F l = 0.042 and 0.084) and one impeller speed (F r = 0.0755). The computational model correctly captured the overall flow field generated by a standard Rushton turbine, including the secondary recirculation loop in the upper part of vessel. The decrease in pumping capacity and power dissipation by impeller with increased aeration was also predicted reasonably well. The model captured the rise in power dissipation for flooding regime in comparison with 3–3 cavity regime. The predicted liquid flow character-

A.R. Khopkar et al. / Chemical Engineering Science 60 (2005) 2215 – 2229

2227

0.2

CARPT data

Fractional number of circulations

0.16

CFD prediction

0.12

0.08

0.04

0

0.0

0.8

1.5

2.3

3.0

4.5

3.8

5.3

6.0

6.8

7.6

t/T c

(a)

0.2

Fractional number of circulations

0.16

0.12

0.08

0.04

0 0.0

0.7

1.3

2.0

2.7

(b)

4.0

3.3

4.7

5.3

6.0

6.7

t/T c

Fig. 10. Comparison of predicted circulation time distribution with experimental data for gas–liquid flows, F r = 0.0755 and Utip = 0.7 m/s: (a) 3–3 cavity regime, F l = 0.042 and F r = 0.0755; and (b) flooding regime, F l = 0.084 and F r = 0.0755.

istics showed reasonable agreement with the CARPT data for both flow regimes. The model, however over-predicted the radial and tangential velocities in the impeller discharge stream. The computational model was then used to study the gas hold-up distribution and circulation time distribution in the vessel. The predicted circulation time distribution was in good comparison with the experimental data and the model captured the increase in circulation time with the increase in gas sparging rate.

Despite some disagreement between the predicted and the experimental results, the computational model shows promising results and seems to be able to predict the essential features of gas–liquid flow for any flow regime. The quasi-steady-state approach used in the present work reasonably captured the Lagrangian information of the flow in the vessel. The model and the results presented in this paper are useful for extending the application of CFD-based models for simulating large multiphase stirred vessels.

2228

A.R. Khopkar et al. / Chemical Engineering Science 60 (2005) 2215 – 2229

Notation B C CD db Di dsp Eo FD g H k N Fl Fr NP NQ Reb P Qg r T t U Utip V x z

research fellowship. CREL authors (ARR, MPD) acknowledge the support of CREL industrial sponsors. impeller blade height, m impeller off-bottom clearance, m drag coefficient bubble diameter, m impeller diameter, m outer diameter of ring sparger, m Evotos number, Eo = g(l − g )db2 /l interphase drag force, N/m3 acceleration due to gravity, m/s2 vessel height, m turbulent kinetic energy, m2 /s2 impeller rotational speed, rps flow number Froude number power number pumping number bubble Reynolds number pressure, N/m2 volumetric gas flow rate, m3 /s radial coordinate, m vessel diameter, m time, s velocity, m/s impeller tip speed, m/s volume of vessel, m3 position vector, m axial coordinate, m

Greek letters


    ϑ

gas volume fraction turbulent kinetic energy dissipation rate, m2 /s3 tangential coordinate Kolmogorov length scale, m viscosity, kg/ms density, kg/m3 effective attenuation function

Subscripts 1 2 q t

liquid gas phase number turbulent

Acknowledgements The authors would like to thank the Department of Science and Technology (India) and the National Science Foundation (NSF) for making this joint research feasible. One of the authors (ARK) is also grateful to CSIR for providing

References Bakker, A., van den Akker, H.E.A., 1994. A computational model for the gas–liquid flow in stirred reactors. Transactions of the IChemE 72, 594–606. Barigou, M., Greaves, M., 1992. Bubble size distribution in a mechanically agitated gas–liquid contactor. Chemical Engineering Science 47 (8), 2009–2025. Brucato, A., Grisafi, F., Montante, G., 1998. Particle drag coefficient in turbulent fluids. Chemical Engineering Science 45, 3295–3314. Buwa, V.V., Ranade, V.V., 2002. Dynamics of gas–liquid flow in rectangular bubble columns. Chemical Engineering Science 57, 4715–4736. Chapman, B.C., Nienow, A.W., Cooke, M., Middleton, J.C., 1983. Particle–gas–liquid mixing in stirred vessels, Part II: Gas–liquid mixing. Chemical Engineering Research and Design 61, 82–95. Chen, P., Sanyal, J., Dudukovic, M.P., 2004. CFD modeling of bubble column flows: implementation of population balance. Chemical Engineering Science 59, 5201–5207. Chen, P., Sanyal, J., Dudukovic, M.P., 2005a. Numerical simulation of bubble column flows: effect of different breakup and coalescence closures. Chemical Engineering Science 60, 1085–1101. Chen, P., Sanyal, J., Dudukovic, M.P., 2005b. Three-dimensional simulation of bubble column flows with bubble coalescence and breakup. A.I.Ch.E. Journal, in press. Deen, N.G., Westerweel, J., Hjertager, B.H., 2001. Upper limit of the gas fraction in PIV measurements in dispersed gas–liquid flows. Proceedings of the Fifth International Conference on Gas–Liquid and Gas–Liquid–Solid Reactor Engineering, Melbourne, Australia. Deen, N.G., Solberg, T., Hjertager, B.H., 2002. Flow generated by an aerated Rushton impeller, two-phase PIV experiments and numerical simulations. Canadian Journal of Chemical Engineering 80 (4), 1–15. Devnathan, N., Moslemian, D., Dudukovic, M.P., 1990. Flow mapping in bubble columns using CARPT. Chemical Engineering Science 45, 2285–2291. Dudukovic, M.P., 2002. Opaque multiphase flows: experiments and modeling. Experimental Thermal and Fluid Science 26 (6–7), 747–761. Gosman, A.D., Lekakou, C., Politis, S., Issa, R.I., Looney, M.K., 1992. Multi-dimensional modeling of turbulent two-phase flows in stirred vessels. A.I.Ch.E. Journal 38 (12), 1947–1956. Hartmann, H., Derksen, J.J., Montavon, C., Pearson, J., Hamill, I.S., van den Akker, H.E.A., 2004. Assessment of large eddy and RANS stirred tank simulations by means of LDA. Chemical Engineering Science 59, 2419–2432. Khopkar, A.R., Aubin, J., Xureb, C., Le Sauze, N., Bertrand, J., Ranade, V.V., 2003. Gas–liquid flow generated by a pitched blade turbine: PIV measurements and CFD simulations. Industrial and Engineering Chemistry and Research 42, 5318–5332. Kumar, S.B., Dudukovic, M.P., 1997. Computer Assisted Gamma and X-ray Tomography: Application to Multiphase Flow, Noninvasive Monitoring of Multiphase Flows. Elsevier, Amsterdam, The Netherlands. Kumar, S.B., Moslemian, D., Dudkovic, M.P., 1995. A gamma ray tomographic scanner for imaging void fraction distribution in bubble columns. Flow Measurement and Instrumentation 6 (1), 61–73. Kumar, S.B., Moslemian, D., Dudukovic, M.P., 1997. Gas holdup measurements in bubble columns using Computed Tomography. A.I.Ch.E. Journal 43 (6), 1414–1425. Lane, G.L., Schwarz, M.P., Evans, G.M., 1999. CFD simulations of gas–liquid flow in a stirred tank. Proceedings of the Third International System on Mixing in Industrial Processes, Osaka, Japan, September 1999, pp. 21–28.

A.R. Khopkar et al. / Chemical Engineering Science 60 (2005) 2215 – 2229 Lane, G.L., Schwarz, M.P., Evans, G.M., 2000. Modeling of the interaction between gas and liquid in stirred vessels. Proceedings of the 10th European Conference on Mixing, pp. 197–204. Morud, K.E., Hjertager, B.H., 1996. LDA measurements and CFD modeling of gas–liquid flow in stirred vessel. Chemical Engineering Science 51, 233–249. Ng, K., Fentiman, N.J., Lee, K.C., Yianneskis, M., 1998. Assessment of sliding mesh CFD predictions and LDA measurements of the flow in a tank stirred by a Rushton impeller. Chemical Engineering Research and Design 76, 737–747. Nienow, A.W., Wisdom, D.J., Middleton, J.C., 1977. The effect of scale and geometry on flooding, recirculation and power in gassed stirred vessels. Paper F1. Proceedings of the European Conference on Mixing, Cambridge. Nienow, A.W., Warmoeskerken, M.M.C.G., Smith, J.M., Konno, M., 1985. On the flooding/loading transition and the complete dispersal condition in aerated vessels agitated by a Rushton turbine, Proceedings of the European Conference on Mixing, Wurzburg, Paper 15. Rafique, M., Chen, P., Dudukovic, M.P., 2004. Computational modeling of gas–liquid flow in bubble columns. Reviews in Chemical Engineering 20 (3–4), 225–375. Rammohan, A.R., 2002. Characterization of single and multiphase flows in stirred tank reactor, Ph.D. Thesis, Washington University. Rammohan, A.R., Kemoun, A., Al-Dahhan, M.H., Dudkovic, M.P., 2001a. A Lagrangian description of flows in stirred tanks via computer automated radioactive particle tracking (CARPT). Chemical Engineering Science 56 (8), 2629–2639. Rammohan, A.R., Kemoun, A., Al-Dahhan, M.H., Dudkovic, M.P., 2001b. Characterization of single phase flows in stirred vessel via computer automated radioactive particle tracking (CARPT). Chemical Engineering Research and Design 79 (A8), 831–844. Rammohan, A.R., Duducovic, M.P., Ranade, V.V., 2003. Eulerian flow field estimation from particle trajectories: numerical experiments for

2229

stirred type flows. Industrial and Engineering Chemistry Research 42, 2589–2601. Ranade, V.V., 1997. An efficient computational model for simulating flow in stirred vessels: a case of Rushton turbine. Chemical Engineering Science 52, 4473–4484. Ranade, V.V., 2002. Computational Flow Modeling for Chemical Reactor Engineering. Academic Press, New York. Ranade, V.V., van den Akker, H.E.A., 1994. A computational snapshot of gas–liquid flow in baffled stirred reactors. Chemical Engineering Science 49, 5175–5192. Ranade, V.V., Deshpande, V.R., 1999. Gas–liquid flow in stirred reactors: trailing vortices and gas accumulation behind impeller blades. Chemical Engineering Science 54, 2305–2315. Ranade, V.V., Perrard, M., Le Sauze, N., Xureb, C., Bertrand, J., 2001a. Trailing vortices of Rushton turbines. Chemical Engineering Research and Design 79A: 3. Ranade, V.V., Perrard, M., Xureb, C., Le Sauze, N., Bertrand, J., 2001b. Influence of gas flow rate on the structure of trailing vortices of a Rushton turbine. Chemical Engineering Research and Design 79A, 957–964. Shah, Y.T., 1992. Design parameters for mechanically agitated reactors. In: Wei, J., Anderson, J.L., Bischoff, K.B. (Eds.), Advances in Chemical Engineering. vol. 17, pp. 1–196. Smith, J.M., 1985. Dispersion of gases in liquids: the hydrodynamics of gas dispersion in low viscosity liquids. In: Ulbrecht, J.J., Patterson, G.K. (Eds.), Mixing of Liquids by Mechanical Agitation. Gordon and Breach Science Publishers, New York, pp. 139–202. Warmoeskerken, M.M.C.G., Smith, J.M., 1985. Flooding of disc turbines in gas–liquid dispersions: a new description of the phenomenon. Chemical Engineering Science 40 (11), 2063–2071. Wechsler, K., Breuer, M., Durst, F., 1999. Steady and unsteady computations of turbulent flows induced by a 4/45◦ pitched blade impeller. Journal of Fluids Engineering 121, 318.